arXiv subjects

Paz Albares

Publications and source records attributed to Paz Albares.

5 recordsLinked to original sources

arXiv2024

Spectral problem for the complex mKdV equation: singular manifold method and Lie symmetries

This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary Darboux transformations and $τ$-functions. The second part concerns the identification of the classical Lie symmetries for the spectral problem. The similarity reductions associated to these symmetries allow us to derive the reduced spectral problems and first integrals for the ordinary differential equations arising from such reductions.

nlin.SI↗

arXiv2021

Miura-reciprocal transformation and symmetries for the spectral problems of KdV and mKdV

We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax pairs for RKdV and RmKdV are straightforwardly obtained by means of the aforementioned reciprocal transformations. We have also identified the classical Lie symmetries for the Lax pairs of RKdV and RmKdV. Non-trivial similarity reductions are computed and they yield non-autonomous ordinary differential equations (ODEs), whose Lax pairs are obtained as a consequence of the reductions.

nlin.SI↗

arXiv2021

Integrability and rational soliton solutions for gauge invariant derivative nonlinear Schrödinger equations

The present work addresses the study and characterization of the integrability of three famous nonlinear Schrödinger equations with derivative-type nonlinearities in 1+1 dimensions. Lax pairs for these three equations are successfully obtained by means of a Miura transformation and the singular manifold method. After implementing the associated binary Darboux transformations, we are able to construct rational soliton-like solutions for those systems.

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arXiv2020

Derivative non-linear Schrödinger equation: Singular manifold method and Lie symmetries

We present a generalized study and characterization of the integrability properties of the derivative non-linear Schrödinger equation in 1+1 dimensions. A Lax pair is derived for this equation by means of a Miura transformation and the singular manifold method. This procedure, together with the Darboux transformations, allow us to construct a wide class of rational soliton-like solutions. Lie classical symmetries have also been computed and similarity reductions have been analyzed and discussed.

nlin.SI↗

arXiv2018

Spectral problem for a two-component nonlinear Schrödinger equation in $2+1$ dimensions: Singular manifold method and Lie point symmetries

An integrable two-component nonlinear Schrödinger equation in $2+1$ dimensions is presented. The singular manifold method is applied in order to obtain a three-component Lax pair. The Lie point symmetries of this Lax pair are calculated in terms of nine arbitrary functions and one arbitrary constant that yield a non-trivial infinite-dimensional Lie algebra. The main non-trivial similarity reductions associated to these symmetries are identified. The spectral parameter of the reduced spectral problem appears as a consequence of one of the symmetries.

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